Russell Gersten

Professor Emeritus, University of Oregon

Director, Instructional Research Group

Russell Gersten

Professor Emeritus, University of Oregon

Director, Instructional Research Group

Needed Future Research on Instructional Practice in

Mathematics

Needed Future Research on Instructional Practice in

Mathematics

“What does not change isthe will to change”

Charles Olson, “The Kingfishers, 1949

There Have Been Changes in Math Instruction Knowledge Base

Small but growing body of empirical research Active sustained engagement by research

mathematicians in the process Insights gained from international

comparisons ….. Though correlational findings hard to interpret accurately

Scores on NAEP grade 4 rising

Response to Intervention (RtI) Increased Interest in Interventions

for Struggling Students

However, problems include:

•Lack of valid screening and progress monitoring measures

•Lack of reliable and valid formative assessment measures

•Need for Interventions at the Secondary Level•Limited connections/misunderstandings between

mathematics education and special education communities

However, problems include:

•Lack of valid screening and progress monitoring measures

•Lack of reliable and valid formative assessment measures

•Need for Interventions at the Secondary Level•Limited connections/misunderstandings between

mathematics education and special education communities

Objectives for the Session

Suggest needed future directions for instructional research in mathematics

Describe troubling conceptual issues that

require advances, if not resolution

Modus Operandi: Build from pockets of strength in the research base.

Sources of Inspiration

• National Mathematics Advisory Panel Report

• Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools

http://www.centeroninstruction.org/

• National Mathematics Advisory Panel Report

• Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools

http://www.centeroninstruction.org/

Sources of Inspiration

• NCTM Curriculum Focal Points

• Meta-analysis of research on teaching students with LD (Gersten, Chard, Jayanthi, Baker, Morphy & Flojo, in press; Review of Educational Research).

• NCTM Curriculum Focal Points

• Meta-analysis of research on teaching students with LD (Gersten, Chard, Jayanthi, Baker, Morphy & Flojo, in press; Review of Educational Research).

Sources of Inspiration

• Research Mathematicians: H. S. Wu, Sybilla Beckman, Jim Lewis, Dick Askey, Jim Milgram, Hy Bass

• Mathematics education: Deborah Ball, Heather Hill, Skip Fennell, Jon Star, Joan Ferrini-Mundy, Jeremy Kilpatrick, Karen Fuson, Bill Schmidt

• Special education/ At risk learners: Anne Foegen, Lynn Fuchs, Brad Witzel, Diane Bryant, Ben Clarke, Asha Jitendra

• Psychology: Dave Geary, Bob Siegler, Bethany Rittle-Johnson

• Research Mathematicians: H. S. Wu, Sybilla Beckman, Jim Lewis, Dick Askey, Jim Milgram, Hy Bass

• Mathematics education: Deborah Ball, Heather Hill, Skip Fennell, Jon Star, Joan Ferrini-Mundy, Jeremy Kilpatrick, Karen Fuson, Bill Schmidt

• Special education/ At risk learners: Anne Foegen, Lynn Fuchs, Brad Witzel, Diane Bryant, Ben Clarke, Asha Jitendra

• Psychology: Dave Geary, Bob Siegler, Bethany Rittle-Johnson

Heavy Lifting:

Major Issues to Broach in Future Research

Instruction1. Teaching fractions and proportions (operations, concepts,

word problems, linkage of number concepts to geometry concepts) so that students understand the mathematics

2. Whole numbers: algorithms and their link to number properties, number lines and number paths, number sense

3. Effective instructional sequences that simultaneously build proficiency in multi-digit multiplication and distributive property of numbers

4. Sequences that integrate word problems with work on procedural fluency

Heavy Lifting

Teacher Knowledge• Content knowledge necessary to teach

grades 3-5, 6,

Measures• Valid screening measures and assessments

for grades 4-8 • Predictors of success in algebra

– Evidence that facility with fractions really does predict success in algebra

Other Interesting Areas1. Role of effort vs. talent in learning

mathematics• Use of this knowledge in interventions for

struggling students

2. Making middle school interventions come alive for students

• What should content be? Integrate with core grade level instruction

• Engagement (Bottge, Woodward, National Research Council)

• Ultimate goal of middle school double dose intervention

Other Interesting Areas

3. Professional development that really makes a difference

4. Exploration of extant data bases (NAEP, state data bases)

• Is mastery of fractions the key?

Where to Start?

1. Findings with some empirical evidence that can serve as a basis for future research

2. Clear areas of need that should serve as a focus for intervention research

3. Areas of fairly broad consensus that require empirical validation or further study

Starting Point: For At-risk Learners/Intervention

Methods

1. Explicit instruction (teachers model easy and hard problems, and think aloud steps in how to solve)

2. Systematic instruction*****

3. Students justify decisions they make

4. Judicious use of concrete representations and consistent use of visual representations

(Source: Gersten, Chard, Jayanthi et al., in press)

1. Explicit instruction (teachers model easy and hard problems, and think aloud steps in how to solve)

2. Systematic instruction*****

3. Students justify decisions they make

4. Judicious use of concrete representations and consistent use of visual representations

(Source: Gersten, Chard, Jayanthi et al., in press)

A Few Starting Points: From Experimental Research

1. Use of contrasting examples (Star)

2. Focus on fluency with combinations/facts (critical for understanding mathematics)

– a mix of practice and work with number families– estimation is more potent– work on fact retrieval is critical

Starting Point: Word Problems

1. Use of common underlying structures to help students figure out how to solve word problems: e.g., change problems (for time), compare problems (for quantity). (Cognitively guided instruction, Jitendra et al, Fuchs et al).

2. Design curricular sequences so that students consistently focus on underlying structure and learn to ignore irrelevant information and translate information from different formats (pictorial, graphs, currency etc.) (Fuchs et al.)

3. Link word problems with procedural examples (Singapore)

A Stab at Putting our Needs and Starting Points Together…

1. Longitudinal research: Is proficiency with fractions and proportions the key to future success in algebra? (extant data bases, new longitudinal research)

2. Word problems and visual representations: extensive use of underlying structure approach and study of comparative impacts of various visual representations?

• Embedded in existing curricula? Asian curricula? • Clarity as to measures of precisely what is taught versus

broader measures of problem solving

A Stab at Putting our Needs and Starting Points Together…

3. Valid screening measures for grades 3-8 for RtI • Are state assessments valid for this purpose?• What is the validity of commonly used benchmark

tests? Reliability?

4. Effective interventions for middle school• Build on what is available and what is known• Continue to use quasi-experiments and descriptive

research to understand impacts of double dose interventions

• Should second mathematics class be integrated with the core class? If so,how?

Professional Development, Teaching and Policy

1. Teaching content knowledge, especially to teachers in grades 3-8

2. Evaluate use of departmentalized mathematics teachers in upper elementary schools

3. Evaluate impact of standards that are challenging but focused and precise. (e.g. Massachusetts, Minnesota, California)

Areas of emerging consensus(National Mathematics Panel)

1. Reciprocal relationship between proficiency with procedures and understanding of mathematical ideas

2. Neither teacher directed nor student-centered learning should be sole instructional means of teaching

3. Topics need much more in depth coverage.

4. At risk learners need some explicit instruction

1. Reciprocal relationship between proficiency with procedures and understanding of mathematical ideas

2. Neither teacher directed nor student-centered learning should be sole instructional means of teaching

3. Topics need much more in depth coverage.

4. At risk learners need some explicit instruction

But We Lack Precision

What do these terms really mean?1. Conceptual understanding of

distributive property: a (b+c) = ab + ac

2. Explicit instruction3. Systematic instruction 4. Guided inquiry5. Use of multiple representations6. Rich, deep mathematical problems

What do these terms really mean?1. Conceptual understanding of

distributive property: a (b+c) = ab + ac

2. Explicit instruction3. Systematic instruction 4. Guided inquiry5. Use of multiple representations6. Rich, deep mathematical problems

And…

Precision is at the heart of mathematics

(Wu, Milgram, and others)

Final Thoughts

• Remember the importance of precise definitions• Remember that there are serious mathematics ideas

that need to be highlighted in texts, state standards, intervention programs:– Equivalence, number line, number properties, linear

functions– Use more operational language to increase precision of our

research and our students’ understanding of mathematics

e.g., minimal use of buzzwords such as conceptual knowledge, real world problems, rich mathematical problems

Final Thought

“ We can be precise”

Charles Olson, “The Kingfishers, 1949